I think Cantor would say you need a proof for that. And I think he would say you can prove it via generating a new real number by going down your set of real numbers and taking the first digit from the first number, the second from the second, third from third, etc. Then you run a transformation on it, for example every number other than 1 becomes 1 and every 1 becomes 2. Then you know that the number you’ve created can’t be first in the set because its first digit doesn’t match, and it can’t be the second number because the second number doesn’t match, etc to infinity. And therefore, if you map your set of whole numbers to your set of real numbers, you’ve discovered a real number that can’t be mapped to a whole number because it can’t be at any position in the set.
Some will say this proves that infinities can be of unequal sizes. Some will more accurately say this shows that uncountable infinities are larger than countable infinities. But the problem I have with it is this: that we begin with the assumption of a set of all real numbers, but then we prove that not all real numbers are contained in the set of all real numbers. We know this because the number we generated literally can not be at any position in the set. This is a paradox. The number is not in the set, therefore we don’t need it to map to a member of the other set. Yet it is a real number and therefore must be in the set. And yet we proved it can’t be in the set.
I’m uncomfortable making inferences based on this type of information. But I’m also not a mathematician. My goal isn’t to start an argument. Maybe somebody who’s better at math can explain it to me better.
I think Cantor would say you need a proof for that. And I think he would say you can prove it via generating a new real number by going down your set of real numbers and taking the first digit from the first number, the second from the second, third from third, etc. Then you run a transformation on it, for example every number other than 1 becomes 1 and every 1 becomes 2. Then you know that the number you’ve created can’t be first in the set because its first digit doesn’t match, and it can’t be the second number because the second number doesn’t match, etc to infinity. And therefore, if you map your set of whole numbers to your set of real numbers, you’ve discovered a real number that can’t be mapped to a whole number because it can’t be at any position in the set.
Some will say this proves that infinities can be of unequal sizes. Some will more accurately say this shows that uncountable infinities are larger than countable infinities. But the problem I have with it is this: that we begin with the assumption of a set of all real numbers, but then we prove that not all real numbers are contained in the set of all real numbers. We know this because the number we generated literally can not be at any position in the set. This is a paradox. The number is not in the set, therefore we don’t need it to map to a member of the other set. Yet it is a real number and therefore must be in the set. And yet we proved it can’t be in the set.
I’m uncomfortable making inferences based on this type of information. But I’m also not a mathematician. My goal isn’t to start an argument. Maybe somebody who’s better at math can explain it to me better.